Universal deformation rings need not be complete intersections. (English) Zbl 1219.11164
Let \(k\) be a perfect field of characteristic \(2\). The authors prove that there is a profinite group \(\Gamma\) and a simple \(k\Gamma\)-module \(V\) such that the universal deformation ring \(R(\Gamma,V)\) is isomorphic to \(W[[t]]/(2t,t^2)\), where \(W=W(k)\) is the ring of infinite Witt vectors over \(k\). In particular, \(R(\Gamma,V)\) is not a complete intersection. There are infinitely many real quadratic fields \(F\) such that one can take the group \(\Gamma\) to be \(\text{Gal}(F^{\text{un}}/F)\), where \(F^{\text{un}}\) is the maximal everywhere unramified extension of \(F\).
Reviewer: Florin Nicolae (Berlin)
Keywords:
profinite group; representation; universal deformation ring; complete intersection; real quadratic field; maximal unramified extensionCitations:
Zbl 1087.11065Software:
PARI/GPReferences:
| [1] | Bleher F.M. (2002). Universal deformation rings and Klein four defect groups. Trans. Am. Math. Soc. 354:3893–3906 · Zbl 1047.20006 · doi:10.1090/S0002-9947-02-03072-6 |
| [2] | Bleher, F.M.: Universal deformation rings and dihedral defect groups, ArXiv math. RT/0607571 · Zbl 1239.20012 |
| [3] | Bleher F.M., Chinburg T. (2000). Universal deformation rings and cyclic blocks. Math. Ann. 318:805–836 · Zbl 0971.20004 · doi:10.1007/s002080000148 |
| [4] | Bleher F.M., Chinburg T. (2003). Applications of versal deformations to Galois theory. Comment. Math. Helv. 78:45–64 · Zbl 1034.20005 |
| [5] | Bleher F.M., Chinburg T. (2006). Universal deformation rings need not be complete intersections. C. R. Math. Acad. Sci. Paris 342: 229–232 · Zbl 1087.11065 |
| [6] | Böckle G. (2001). On the density of modular points in universal deformation spaces. Amer. J. Math. 123:985–1007 · Zbl 0984.11025 · doi:10.1353/ajm.2001.0031 |
| [7] | Böckle G., Khare C. (2006). Mod l representations of arithmetic fundamental groups. II. A conjecture of A. J. de Jong. Compos. Math. 142:271–294 · Zbl 1186.11029 · doi:10.1112/S0010437X05002022 |
| [8] | Boston, N.: Galois p-groups unramified at p–a survey. In: Primes and Knots (eds.) Toshitake Kohno and Masanori Morishita, Contemp. Math. (to appear) · Zbl 1188.11057 |
| [9] | Chinburg, T.: Can deformation rings of group representations not be local complete intersections? In: Cornelissen, G., Oort, F. (eds) Bouw, I., Chinburg, T., Cornelissen, C.G., Glass, D., Lehr, C., Matignon, M., Oort, R.P., Wewers, S., Problems from the Workshop on Automorphisms of Curves. Rend. Sem. Mat. Univ. Padova 113, 129–177 (2005) |
| [10] | de Jong A.J. (2001). A conjecture on arithmetic fundamental groups. Israel J. Math. 121:61–84 · Zbl 1054.11032 · doi:10.1007/BF02802496 |
| [11] | de Smit, B., Lenstra, H.W.: Explicit construction of universal deformation rings. In: Modular Forms and Fermat’s Last Theorem, pp. 313–326 (Boston, MA, 1995). Springer, Berlin Heidelberg New York, (1997) · Zbl 0907.13010 |
| [12] | Emerton M., Calegari F. (2005). On the ramification of Hecke algebras at Eisenstein primes. Invent. Math. 160: 97–144 · Zbl 1145.11314 · doi:10.1007/s00222-004-0406-z |
| [13] | Emerton M., Pollack R., Weston T. (2006). Variation of Iwasawa invariants in Hida families. Invent. Math. 163:523–580 · Zbl 1093.11065 · doi:10.1007/s00222-005-0467-7 |
| [14] | Gaitsgory, D.: On de Jong’s conjecture, ArXiv math.AG/0402184. |
| [15] | Grothendieck A. (1967). Éléments de géométrie algébrique, Chapitre IV, Quatriéme Partie. Publ. Math. IHES 32:5–361 |
| [16] | Kisin M. (2003). Overconvergent modular forms and the Fontaine–Mazur conjecture. Invent. Math. 153:373–454 · Zbl 1045.11029 · doi:10.1007/s00222-003-0293-8 |
| [17] | Kisin, M.: Moduli of finite flat group schemes and modularity. Preprint (2006) · Zbl 1201.14034 |
| [18] | Mazur, B.: Deforming Galois representations. In: Galois groups over \(\mathbb{Q}\) , pp. 385–437 (Berkeley, CA, 1987). Springer, Berlin Heidelberg New York (1989) |
| [19] | PARI2, PARI/GP, version 2.1.5, Bordeaux (2004) megrez.math.u-bordeaux.fr/pub/numberfields/ |
| [20] | Serre, J.P.: Modular forms of weight one and Galois representations. In: Algebraic number fields: L-functions and Galois properties. Proc. Sympos., Univ. Durham, Durham, 1975, 193–268. Academic, London (1977) |
| [21] | Taylor R., Wiles A. (1995). Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141:553–572 · Zbl 0823.11030 · doi:10.2307/2118560 |
| [22] | Washington L.C. (1982). Introduction to Cyclotomic Fields. Springer, Berlin Heidelberg New York · Zbl 0484.12001 |
| [23] | Wiles A. (1995). Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141:443–551 · Zbl 0823.11029 · doi:10.2307/2118559 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
